direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40⋊C2, C8⋊8D10, C4.6D20, C40⋊9C22, C10⋊1SD16, C20.29D4, C20.28C23, D20.6C22, C22.12D20, Dic10⋊3C22, (C2×C8)⋊5D5, (C2×C40)⋊7C2, C5⋊1(C2×SD16), C10.9(C2×D4), (C2×D20).5C2, C2.11(C2×D20), (C2×C10).16D4, (C2×C4).79D10, (C2×Dic10)⋊5C2, C4.26(C22×D5), (C2×C20).88C22, SmallGroup(160,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C40⋊C2
G = < a,b,c | a2=b40=c2=1, ab=ba, ac=ca, cbc=b19 >
Subgroups: 280 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C2×SD16, C40 [×2], Dic10 [×2], Dic10, D20 [×2], D20, C2×Dic5, C2×C20, C22×D5, C40⋊C2 [×4], C2×C40, C2×Dic10, C2×D20, C2×C40⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, D10 [×3], C2×SD16, D20 [×2], C22×D5, C40⋊C2 [×2], C2×D20, C2×C40⋊C2
(1 77)(2 78)(3 79)(4 80)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 77)(2 56)(3 75)(4 54)(5 73)(6 52)(7 71)(8 50)(9 69)(10 48)(11 67)(12 46)(13 65)(14 44)(15 63)(16 42)(17 61)(18 80)(19 59)(20 78)(21 57)(22 76)(23 55)(24 74)(25 53)(26 72)(27 51)(28 70)(29 49)(30 68)(31 47)(32 66)(33 45)(34 64)(35 43)(36 62)(37 41)(38 60)(39 79)(40 58)
G:=sub<Sym(80)| (1,77)(2,78)(3,79)(4,80)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,77)(2,56)(3,75)(4,54)(5,73)(6,52)(7,71)(8,50)(9,69)(10,48)(11,67)(12,46)(13,65)(14,44)(15,63)(16,42)(17,61)(18,80)(19,59)(20,78)(21,57)(22,76)(23,55)(24,74)(25,53)(26,72)(27,51)(28,70)(29,49)(30,68)(31,47)(32,66)(33,45)(34,64)(35,43)(36,62)(37,41)(38,60)(39,79)(40,58)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,77)(2,56)(3,75)(4,54)(5,73)(6,52)(7,71)(8,50)(9,69)(10,48)(11,67)(12,46)(13,65)(14,44)(15,63)(16,42)(17,61)(18,80)(19,59)(20,78)(21,57)(22,76)(23,55)(24,74)(25,53)(26,72)(27,51)(28,70)(29,49)(30,68)(31,47)(32,66)(33,45)(34,64)(35,43)(36,62)(37,41)(38,60)(39,79)(40,58) );
G=PermutationGroup([(1,77),(2,78),(3,79),(4,80),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,77),(2,56),(3,75),(4,54),(5,73),(6,52),(7,71),(8,50),(9,69),(10,48),(11,67),(12,46),(13,65),(14,44),(15,63),(16,42),(17,61),(18,80),(19,59),(20,78),(21,57),(22,76),(23,55),(24,74),(25,53),(26,72),(27,51),(28,70),(29,49),(30,68),(31,47),(32,66),(33,45),(34,64),(35,43),(36,62),(37,41),(38,60),(39,79),(40,58)])
C2×C40⋊C2 is a maximal subgroup of
C8⋊5D20 C8.8D20 C42.16D10 C8⋊D20 C8.D20 D20.31D4 D20.32D4 D20⋊14D4 Dic10⋊14D4 Dic10⋊2D4 D20.8D4 D4⋊3D20 D20.D4 Dic10.11D4 Q8⋊2D20 Q8.D20 Dic5⋊SD16 C20⋊SD16 D20.19D4 C42.36D10 Dic10⋊8D4 Dic5⋊8SD16 C8⋊8D20 C8⋊3D20 C40⋊21(C2×C4) C8.24D20 C40⋊30D4 C40⋊2D4 D4.3D20 C40⋊11D4 C40.43D4 C40⋊15D4 C40.37D4 D4.11D20 C2×D5×SD16 D8⋊11D10
C2×C40⋊C2 is a maximal quotient of
C40⋊9Q8 C20.14Q16 C8⋊5D20 C4.5D40 C23.34D20 D20.31D4 C23.38D20 Dic10⋊14D4 C20⋊SD16 D20⋊3Q8 Dic10⋊8D4 Dic10⋊4Q8 C40⋊30D4
46 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
type | + | + | + | + | + | + | + | + | + | + | + | + | ||
image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | D10 | D10 | D20 | D20 | C40⋊C2 |
kernel | C2×C40⋊C2 | C40⋊C2 | C2×C40 | C2×Dic10 | C2×D20 | C20 | C2×C10 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
# reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C2×C40⋊C2 ►in GL3(𝔽41) generated by
40 | 0 | 0 |
0 | 1 | 0 |
0 | 0 | 1 |
1 | 0 | 0 |
0 | 27 | 28 |
0 | 13 | 18 |
1 | 0 | 0 |
0 | 1 | 34 |
0 | 0 | 40 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,27,13,0,28,18],[1,0,0,0,1,0,0,34,40] >;
C2×C40⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_{40}\rtimes C_2
% in TeX
G:=Group("C2xC40:C2");
// GroupNames label
G:=SmallGroup(160,123);
// by ID
G=gap.SmallGroup(160,123);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,50,579,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^2=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations